Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise

Amplitude equations for SPDEs with quadratic nonlinearities forced by additive and multiplicative noise

This article deals with stochastic partial differential equations with quadratic nonlinearities perturbed by small additive and multiplicative noise. We present the approximate solution of the original equation via the amplitude equation and give the rigorous error analysis. For illustration, we apply our main theorems to stochastic Burger's equation.

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone coefficients such as stochastic Burgers equations with additive trace-class noise. The key idea in the proof of our main result is (i) to bring the classical Alekseev-Gröbner formula from deterministic analysis into play and (ii) to employ uniform exponential moment estimates for the numerical approximations.

由乘性Rosenblatt噪声驱动的随机Burgers方程：局部存在性、唯一性和正则性

Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity

本文研究由乘性Rosenblatt噪声驱动的随机Burgers方程，证明了弱解的局部存在性与唯一性，并建立了解的Hölder正则性。

关于随机Burgers型方程收敛率的最优结果

Optimal rate of convergence for stochastic Burgers-type equations

本文研究了加性噪声下随机Burgers型方程的数值解收敛性，证明了其收敛率可接近1/2，改进了之前关于收敛率接近1/6的结果。

随机受迫 Korteweg-de Vries-Burgers 方程的遍历性

Ergodicity for the randomly forced Korteweg-de Vries-Burgers equation

本文研究非可加白噪声驱动的 KdVB 方程遍历性，利用新解析性质与耦合及渐近耦合方法，证明局部化噪声下的指数遍历性和乘性噪声下的遍历性。

Predictability of the Burgers dynamics under model uncertainty

Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random boundary forcing or with random body forcing. Our goal is to attempt to understand the stochastic Burgers dynamics by predicting or estimating the solution processes in various diagnostic metrics, such as mean length scale, correlation function and mean energy. First, for the linearized model, we observe that the important statistical quantities like mean ene…

关于具有分数型噪声的随机Burgers方程的全离散方案的强收敛性

Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise

本文研究了由加性圆柱分数布朗运动驱动的随机Burgers方程的数值近似，采用谱伽辽金方法空间离散，结合非线性 tame 加速指数欧拉方法构建全离散方案，通过证明分数布朗运动的随机卷积指数可积性，得出半离散和全离散近似的矩有界性，并利用停止时间技术证明方案的强收敛性。

粗糙的Burgers型随机偏微分方程

Rough Burger-like SPDEs

本文提出了一种在低正则性区域下，通过改进的粗糙路径方法，建立Burgers型随机偏微分方程路径存在性和唯一性的理论框架。

Galerkin近似用于随机Burgers方程

Galerkin approximations for the stochastic Burgers equation

本文研究了随机偏微分方程中Galerkin近似的存在性、唯一性和收敛性，特别针对随机Burgers方程，提出了在L^∞拓扑下统一估计的新型方法。

广义分数奇 Burgers 方程的适定性研究：驱动由 $|D|^{\frac{1}{2}}ξ$

A Generalized Framework for Singular Fractional Burgers Equations with Stochastic Forcing

本文研究了由 $|D|^{1/2}ξ$ 驱动的广义分数奇 Burgers 方程的适定性，建立了 GFSB 方程框架并证明了其局部适定性，以及在特定条件下解的广义性质。

Deviation bounds for additive functionals of Markov process

In this paper we derive non asymptotic deviation bounds for $$¶_ν(|\frac 1t \int_0^t V(X_s) ds - \int V dμ| \geq R)$$ where $X$ is a $μ$ stationary and ergodic Markov process and $V$ is some $μ$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $μ$. Regularity means here that $μ$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc...).

Hairer-Quastel universality in non-stationarity via energy solution theory

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory Goncalves-Jara '14, Gubinelli-Jara '13, Gubinelli-Perkowski '18, Gubinelli-Perkowski '20. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution Gubinelli-Perkowski '17 plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, fir…

Distribution dependent SDEs driven by additive continuous noise

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

Accessibility of SPDEs driven by pure jump noise and its applications

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of si…

Singularity Formation in the Deterministic and Stochastic Fractional Burgers Equation

This study is motivated by the question of how singularity formation and other forms of extreme behavior in nonlinear dissipative partial differential equations are affected by stochastic excitations. To address this question we consider the 1D fractional Burgers equation with additive colored noise as a model problem. This system is interesting, because in the deterministic setting it exhibits finite-time blow-up or a globally well-posed behavior depending on the value of the fractional dissipation exponent. The problem is studied by performing a series of accurate numerical computations comb…

Geometric stochastic heat equations

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\mathbf{R}^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties: - For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously …

Multiscale Analysis for SPDEs with Quadratic Nonlinearities

In this article we derive rigorously amplitude equations for stochastic PDEs with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one dimensional Burgers equation for…

Random attractors for locally monotone stochastic partial differential equations

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$α$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic porous media equations and stochastic $p$-Laplace equations.

Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient

We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [16]. The tools we develop are very general and can be used to study many other examples of applications.

Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on ${\mathbb{S}}^2$

We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

Malliavin regularity and weak approximation of semilinear SPDE with Lévy noise

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

关于局部单调随机演化方程的遍历性与混合性

Ergodicity and mixing for locally monotone stochastic evolution equations

本文研究了局部单调驱动和退化加性Wiener噪声的随机演化方程，建立了存在唯一不变概率测度的定量条件，并提供了解的矩估计和半群的e-性质，以及2-Wassersteinε-混合时间的上界。

Stochastic Differential Equations with Critical Drifts

We establish the well-posedness of SDE with the additive noise when a singular drift belongs to the critical spaces. We prove that if the drift belongs to the Orlicz-critical space $L^{q,1}([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p} =1$, then the corresponding SDE admits a unique strong solution. We also derive the Sobolev regularity of a solution under the Orlicz-critical condition.

The stochastic $p$-Laplace equation on $\mathbb{R}^d$

We show well-posedness of the $p$-Laplace evolution equation on $\mathbb{R}^d$ with square integrable random initial data for arbitrary $1<p<\infty$ and arbitrary space dimension $d\in\mathbb{N}$. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the $p$-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of $1<p<\infty$ and also depends on the space dimension $d$. We propose a framework of functional space…

从随机粒子系统微观推导具有反常扩散的非局部模型

Microscopic derivation of non-local models with anomalous diffusions from stochastic particle systems

本文从由Lévy过程驱动的相互作用粒子系统出发，在弱正则性条件下推导出一类含反常扩散和非局部奇异核的非线性积分-微分方程，并给出微观经验测度与极限方程解在非齐次Sobolev空间中的定量估计。

Random attractor of the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under $non$-$periodic$ boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the uniform $a$ $priori$ estimates in some Sobolev space whose regularity is high enough. But this is very complicated for the 3D stochastic PEs with the $non$-$periodic$ boundary conditions. Therefore…

Stochastic nonlinear Schrödinger equations on tori

We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.

Non-uniqueness in law of transport-diffusion equation forced by random noise

We consider a transport-diffusion equation forced by random noise of three types: additive, linear multiplicative in It$\hat{\mathrm{o}}$'s interpretation, and transport in Stratonovich's interpretation. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport-diffusion equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.

量化Gyöngy与Krylov的一个收敛定理

Quantifying a convergence theorem of Gyöngy and Krylov

本文量化了Gyöngy与Krylov的收敛定理，推导出在无漂移系数正则性条件下Euler-Maruyama格式逼近多维SDE的精确强收敛率，并证明加性噪声下利用Sobolev正则性可获改进速率。

阻尼随机Korteweg-de Vries方程的不变测度的一致时间逼近与收敛性

Uniform-in-time approximation and convergence of invariant measuresfor the damped stochastic Korteweg-de Vries equation

针对加性噪声驱动的周期阻尼sKdV方程，提出Lie-Trotter分裂逼近，在足够大阻尼下证明强、弱阶1的一致时间收敛及不变测度的阶1逼近，为长时间模拟提供途径。

Stochastic logarithmic Schrodinger equations: energy regularized approach

In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schrödinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schrödinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal Hölder regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_α(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equati…

关于具有局部单调系数的SPDE的平稳解和不变测度的大偏差原理

Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients

本文研究了具有局部单调系数的SPDE的平稳解的存在性，并证明了这些解的大偏差原理，通过收缩原理推导出不变测度的大偏差原理，避免了构造准势能和验证Dembo-Zeitouni均匀大偏差原理的需要。

高温区相互耦合振子异常行为的推导

Derivation of anomalous behavior from interacting oscillators in the high-temperature regime

本文在高温区研究具有体积和能量守恒的相互耦合振子模型，通过泰勒展开提取谐振主部，并证明两个涨落场分别收敛于 Ornstein-Uhlenbeck 过程或随机 Burgers 方程，以及偏斜 Lévy 过程驱动的分数阶扩散方程。

Parameter estimation for semilinear SPDEs from local measurements

This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in Altmeyer, Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs that satisfy some abstract, and verifiable, conditions. The proofs of asymptotic results are based on splitting the solution in linear and nonlinear parts and fine regularity properties in $L^p$-spaces. The ob…

实直线上随机Korteweg - de Vries方程的无条件适定性

Unconditional well-posedness of the stochastic Korteweg-de Vries equation on the real line

研究实直线上带加性噪声的随机Korteweg - de Vries方程的适定性，用傅里叶限制范数方法证其在\(L^2(\mathbb{R})\)中的全局适定性，还通过改进论证证明了最优路径无条件唯一性，附录给出相关有界性论证。

具有分数阶Sobolev正则漂移系数的随机微分方程强逼近的最优误差率

On optimal error rates for strong approximation of SDEs with a drift coefficient of fractional Sobolev regularity

研究漂移系数具有分数阶Sobolev正则性的标量加性噪声SDE的强逼近，证明对于s∈(1/2,1)，等距Euler逼近的L²误差率(1+s)/2在一般情形下不可改进，并给出匹配下界。

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations I: The Stochastic Semiflow

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this paper), we prove a general existence and compactness theorem for $C^k$-cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with …

带Hölder系数的随机热方程的指数遍历性

Exponential ergodicity of stochastic heat equations with Hölder coefficients

该研究针对带仅Hölder连续漂移与扩散系数的时空白噪声驱动随机热方程，首次证明特定参数下依分布唯一温和解的存在性，还在Lyapunov函数假设下证得解指数收敛到唯一不变测度，且技术可推广到Burgers型非线性SPDE情形。

Global dynamics for the two-dimensional stochastic nonlinear wave equations

We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing c…

Normal approximation of the solution to the stochastic heat equation with Lévy noise

Given a sequence $\dot{L}^{\varepsilon}$ of Lévy noises, we derive necessary and sufficient conditions in terms of their variances $σ^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise $σ(\varepsilon)^{-1} \dot{L}^\varepsilon$ converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of S. Asmussen and J. Rosiński [J. Appl. Probab. 38 (2001) 482-493] and S. Cohen and J. Rosiński [Bernoulli 13 (2007) 195-210] for finite-dimensional Lévy processes…

Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise

The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the litera…

Non-uniqueness in law of three-dimensional Navier-Stokes equations diffused via a fractional Laplacian with power less than one half

Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hop…

带超临界漂移的非局部算子的哈纳克不等式及其应用

Harnack inequalities for nonlocal operators with supercritical drifts and their applications

本文研究带临界/超临界无散度漂移的非局部方程弱解的哈纳克估计，应用该结果建立了三类方程（临界随机准地转、二维分数阶纳维-斯托克斯、临界随机微分方程广义鞅问题）的适定性或弱解存在性。

Path properties of the solution to the stochastic heat equation with Lévy noise

We consider sample path properties of the solution to the stochastic heat equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a Lévy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than $-\frac d 2$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Lévy noise such that noises with a smaller ind…

Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains

In this work, we consider the two and three-dimensional stochastic convective Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive white noise $$\mathrm{d}\boldsymbol{u}-[μΔ\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-α\boldsymbol{u}-β|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla p]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\mathrm{d}\mathrm{W},\ \nabla\cdot\boldsymbol{u}=0,$$ for $r\in[1,\infty),$ $μ,α,β>0$ in unbounded domains (like Poincaré domains) $\mathcal{O}\subset\mathbb{R}^d$ ($d=2,3$) where $\mathrm{W}(\cdot)$ is a Hilbert space valued Wiener process on…

A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide …

受扰KPZ的高斯结构

The Gaussian structure of a perturbed KPZ

研究带加性空间扰动的KPZ方程，证明其存在唯一不变测度，该测度关于布朗桥绝对连续，且在扰动较小时具有有限相对熵。

无界域中Landau-Lifshitz-Bloch方程的不变测度及其极限行为

Invariant measures and their limiting behavior of the Landau-Lifshitz-Bloch equation in unbounded domains

本文研究无界域上带噪声的随机Landau-Lifshitz-Bloch方程的不变测度，通过区域扩张法证解的存在唯一性，用弱Feller论证得不变测度存在性，一维情形下证不变测度集一致胎紧且扰动方程不变测度的极限为极限系统的不变测度，发展了克服无界域困难的相关技术。